1) By Theorem 4.2.4 of the reference below, the set $D$ is thin at $z\in\partial D$ if and only if $z$ is an irregular point of the complement $D^{c}$. Moreover, by Theorem 4.2.5, the set of irregular boundary points of a domain is polar.

2) Let $D$ be the union of the unit disk $\mathbb{D}$ and the segment $E=[2,3]$. Then, $\overline D\setminus E=\mathbb{D}$ is thin at each point of $E$ and $E$ is not polar.

> T. Ransford, Potential theory in the complex plane, Cambridge, 1995.